This book is a collection of studies conducted by a team of researchers to better understand some of the important elements of effective teaching of mathematics in middle grades. The studies were conducted over a period of six years and were led by Gerald Kulm. The authors identified curriculum, content knowledge (CK), pedagogical content knowledge (PCK), professional development, ongoing support and instructional materials as the key factors that influence the teaching and learning of mathematics. The studies involved collecting and analyzing data using statistical methods to gain insight into how these factors come together to enhance student learning of mathematics.
Depending on the study, data were collected from pre-service teachers, teachers with some teaching experience, or a diverse population of students in middle grades. Each study gathered data using one or more of the following methods: tests on CK and PCK, interviews, surveys, video tapes and pre-test and post-test data . Throughout the book the authors have referenced and quoted results from a wide range of articles to keep the reader abreast of what is known in the areas they are investigating and how their research questions are either different from, or build on, existing studies.
A couple of the studies involved investigation of CK and PCK of pre-service teachers on the topics of linear functions and probability and statistics. These studies were very interesting because the authors included samples of short answer and multiple choice questions from tests used to gather data, along with samples of student responses which were analyzed both statistically and qualitatively. In the case of linear functions, it was particularly interesting to see two sets of questions on the same given information, one set to test content knowledge and the other to test pedagogical content knowledge. The questions tested understanding of different forms of visual representation such as graphs, and tables and their flexibility to translate within visual representations and between visual and symbolic representations. The analysis indicated that this kind of flexibility is important for improving student learning. In the case of probability and statistics, the study showed a gap between understanding and explaining a concept, indicating a need for deeper understanding. These and other studies highlight the impact of CK in mathematics on PCK.
Teachers’ CK and PCK of fractions, another important topic in middle grades, has been investigated in at least two of the studies. Video tapes of 6th grade teachers were the main source of data. Even though the sample size was small, detailed excerpts from the video tapes and interviews provided insight into the lack of consistency on the part of teachers in using a variety of tools to explain the concept of fractions and identify and correct student errors or misconceptions about equivalent fractions.
In one study pre-test and post-test data from a large number of middle grades students who had algebra for one year were analyzed to understand why some misconceptions are resistant to change and it was found that sometimes students find it difficult to make the transformation from a process-oriented to an object-oriented level. Studies on how professional development with appropriate curriculum materials help teachers’ to improve their content knowledge is also presented.
There are many interesting instances of actual classroom teaching by pre-service teachers and teachers with some teaching experience. These instances make one think about how to stimulate student thinking and promote understanding. Therefore parts of this book will be useful to pre-service teachers, novice middle school teachers, and teachers with some experience seeking to improve student learning. The authors have also used a variety of statistical methods to compare different elements of teaching and have given a detailed account of it in various chapters. This book could therefore also be appealing as a reference to college professors who teach methods courses in mathematics.
Hema Gopalakrishnan is associate professor of mathematics at Sacred Heart University in Fairfield, CT.